SRC/dbdsqr.f(3)

NAME
SRC/dbdsqr.f

SYNOPSIS

  Functions/Subroutines
subroutine dbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info)
DBDSQR

Function/Subroutine Documentation

  subroutine dbdsqr (character uplo, integer n, integer ncvt, integer nru, integer ncc, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldvt, * ) vt, integer ldvt, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer info)

DBDSQR

Purpose:

    !>
    !> DBDSQR computes the singular values and, optionally, the right and/or
    !> left singular vectors from the singular value decomposition (SVD) of
    !> a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
    !> zero-shift QR algorithm.  The SVD of B has the form
    !>
    !>    B = Q * S * P**T
    !>
    !> where S is the diagonal matrix of singular values, Q is an orthogonal
    !> matrix of left singular vectors, and P is an orthogonal matrix of
    !> right singular vectors.  If left singular vectors are requested, this
    !> subroutine actually returns U*Q instead of Q, and, if right singular
    !> vectors are requested, this subroutine returns P**T*VT instead of
    !> P**T, for given real input matrices U and VT.  When U and VT are the
    !> orthogonal matrices that reduce a general matrix A to bidiagonal
    !> form:  A = U*B*VT, as computed by DGEBRD, then
    !>
    !>    A = (U*Q) * S * (P**T*VT)
    !>
    !> is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
    !> for a given real input matrix C.
    !>
    !> See  by J. Demmel and W. Kahan,
    !> LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
    !> no. 5, pp. 873-912, Sept 1990) and
    !>  by
    !> B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
    !> Department, University of California at Berkeley, July 1992
    !> for a detailed description of the algorithm.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  B is upper bidiagonal;
    !>          = 'L':  B is lower bidiagonal.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix B.  N >= 0.
    !>

    NCVT

    !>          NCVT is INTEGER
    !>          The number of columns of the matrix VT. NCVT >= 0.
    !>

    NRU

    !>          NRU is INTEGER
    !>          The number of rows of the matrix U. NRU >= 0.
    !>

    NCC

    !>          NCC is INTEGER
    !>          The number of columns of the matrix C. NCC >= 0.
    !>

    D

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          On entry, the n diagonal elements of the bidiagonal matrix B.
    !>          On exit, if INFO=0, the singular values of B in decreasing
    !>          order.
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          On entry, the N-1 offdiagonal elements of the bidiagonal
    !>          matrix B.
    !>          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
    !>          will contain the diagonal and superdiagonal elements of a
    !>          bidiagonal matrix orthogonally equivalent to the one given
    !>          as input.
    !>

    VT

    !>          VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
    !>          On entry, an N-by-NCVT matrix VT.
    !>          On exit, VT is overwritten by P**T * VT.
    !>          Not referenced if NCVT = 0.
    !>

    LDVT

    !>          LDVT is INTEGER
    !>          The leading dimension of the array VT.
    !>          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
    !>

    U

    !>          U is DOUBLE PRECISION array, dimension (LDU, N)
    !>          On entry, an NRU-by-N matrix U.
    !>          On exit, U is overwritten by U * Q.
    !>          Not referenced if NRU = 0.
    !>

    LDU

    !>          LDU is INTEGER
    !>          The leading dimension of the array U.  LDU >= max(1,NRU).
    !>

    C

    !>          C is DOUBLE PRECISION array, dimension (LDC, NCC)
    !>          On entry, an N-by-NCC matrix C.
    !>          On exit, C is overwritten by Q**T * C.
    !>          Not referenced if NCC = 0.
    !>

    LDC

    !>          LDC is INTEGER
    !>          The leading dimension of the array C.
    !>          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (LWORK)
    !>          LWORK = 4*N, if NCVT = NRU = NCC = 0, and
    !>          LWORK = 4*(N-1), otherwise
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  If INFO = -i, the i-th argument had an illegal value
    !>          > 0:
    !>             if NCVT = NRU = NCC = 0,
    !>                = 1, a split was marked by a positive value in E
    !>                = 2, current block of Z not diagonalized after 30*N
    !>                     iterations (in inner while loop)
    !>                = 3, termination criterion of outer while loop not met
    !>                     (program created more than N unreduced blocks)
    !>             else NCVT = NRU = NCC = 0,
    !>                   the algorithm did not converge; D and E contain the
    !>                   elements of a bidiagonal matrix which is orthogonally
    !>                   similar to the input matrix B;  if INFO = i, i
    !>                   elements of E have not converged to zero.
    !>

Internal Parameters:

    !>  TOLMUL  DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
    !>          TOLMUL controls the convergence criterion of the QR loop.
    !>          If it is positive, TOLMUL*EPS is the desired relative
    !>             precision in the computed singular values.
    !>          If it is negative, abs(TOLMUL*EPS*sigma_max) is the
    !>             desired absolute accuracy in the computed singular
    !>             values (corresponds to relative accuracy
    !>             abs(TOLMUL*EPS) in the largest singular value.
    !>          abs(TOLMUL) should be between 1 and 1/EPS, and preferably
    !>             between 10 (for fast convergence) and .1/EPS
    !>             (for there to be some accuracy in the results).
    !>          Default is to lose at either one eighth or 2 of the
    !>             available decimal digits in each computed singular value
    !>             (whichever is smaller).
    !>
    !>  MAXITR  INTEGER, default = 6
    !>          MAXITR controls the maximum number of passes of the
    !>          algorithm through its inner loop. The algorithms stops
    !>          (and so fails to converge) if the number of passes
    !>          through the inner loop exceeds MAXITR*N**2.
    !>
    !>

Note:

    !>  Bug report from Cezary Dendek.
    !>  On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
    !>  removed since it can overflow pretty easily (for N larger or equal
    !>  than 18,919). We instead use MAXITDIVN = MAXITR*N.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 241 of file dbdsqr.f.

Author

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